## Abstract

We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an ω-narrow topological group G: (i)G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

Original language | English |
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Pages (from-to) | 5645-5664 |

Number of pages | 20 |

Journal | Transactions of the American Mathematical Society |

Volume | 369 |

Issue number | 8 |

DOIs | |

State | Published - 1 Jan 2017 |

## Keywords

- Locally compact group
- Pro-Lie group
- Separable topological space
- Topological group

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics